Minimally Invasive Procedures (MIPs) emerged as an alternative to more invasive surgical approaches, offering patient benefits such as smaller incisions, less pain, and shorter hospital stay. In one class of MIPs, where natural body lumens or small incisions are used to access deeper anatomical locations, Flexible Surgical and Interventional Robots (FSIRs) such as catheters and endoscopes are widely used. Due to their flexible and compliant nature, FSIRs can be inserted via natural orifices or small incisions, then moved towards hard-to-reach targets to perform interventional tasks. However, existing FSIRs are confronted with challenges in sensing, control, and navigation. These issues stem from the robot's non-linear behavior and the intricate nature of the lumens, where accurately modeling the complex interactions and disturbances proves to be exceptionally difficult. The rapid advances in Machine Learning (ML) have facilitated the widespread adoption of ML techniques in FSIRs. This article provides an overview of these efforts by first introducing a classification of existing ML algorithms, including traditional ML methods and modern Deep Learning (DL) approaches, commonly used in FSIRs. Next, the use of ML algorithms is surveyed per sub-domain, namely for perception, modeling, control, and navigation. Trends, popularity, strengths, and/or limitations of different ML algorithms are analyzed. The different roles that ML plays among tasks are investigated and described. Finally, discussions are conducted on the limitations and the prospects of ML in MIPs.